Planes in Space

This section expects students to have familiarity with the notation and operations of vectors from . We also would like to point to our earlier description of flat objects in space to motivate the difference between lines and planes as our flat objects. The examples and activities in this section are selected to highlight the broad range of ways that information about a plane can be given, including talking about parallel planes, the interaction between equations of planes and lines, and the angle between planes.

This section can be covered in a longer class period alongside Planes. Many exercises in this section include taking information from planes to get information about a related line and vice versa, so you may want to select work from the sections on lines and planes together.

In , we saw how to describe a line in \(\R^n\) by setting an initial point and allowing unrestricted movement in a direction given by a direction vector. In this section's Preview Activity, we will be looking at what happens when we allow only movement that is perpendicular to a given direction. Since we will specify the given direction by a vector, we will find the perpendicular directions using orthogonal vectors.

Now that we have a way of describing lines, we would like to develop a means of describing planes in three dimensions. In Section, we studied the coordinate planes and planes parallel to them. In particular, \(x=1\), \(y=-2\), and \(z=\sqrt{3}\) are examples of fundamental planes. In general, the equations of fundamental planes have the form \(coordinate = constant\).

As shown in , any vector in a plane with \(x= \text{constant}\) will be orthogonal to the vector \(\langle 1,0,0 \rangle\), any vector in a plane with \(y= \text{constant}\) will be orthogonal to the vector \(\langle 0,1,0 \rangle\), and any vector in a plane with \(z= \text{constant}\) will be orthogonal to the vector \(\langle 0,0,1 \rangle\). We will use this idea to define a plane in general.

Like the definition of a line, the definition of a plane given above uses a starting point and a vector as the critical pieces of information. For a line, you begin at the starting point and move as much as you want parallel to the given vector (the direction vector). For a plane, you begin at the starting point and move as much as you want orthogonal to the given vector (the normal vector). For a line, you move only in the direction of the given vector whereas on a plane you cannot move at all in the direction of the given vector.

This description of a plane allows us to find the equation of a plane. Assume that \(\vn=\langle a,b,c\rangle\), \(P_0 = (x_0, y_0, z_0)\), and that \(Q=(x,y,z)\) is an arbitrary point on the plane. Since the vector \(\overrightarrow{P_0 Q}\) lies in the plane, it must be perpendicular to \(\vn\). This means that \[\begin{aligned}0 =\mathstrut \amp \vn\cdot\overrightarrow{P_0 Q} \\ =\mathstrut \amp \vn\cdot \big(\langle x,y,z \rangle - \langle x_0, y_0, z_0\rangle\big) \\ =\mathstrut \amp \vn \cdot \langle x-x_0, y-y_0, z-z_0 \rangle \\ =\mathstrut \amp a(x-x_0) + b(y-y_0) + c(z-z_0).\end{aligned}\]

We may now summarize our new equation for a plane.

• The scalar equation of the plane with normal vector \(\vn =\langle a,b,c \rangle\) containing the point \(P_0 = (x_0, y_0, z_0)\) is \[\begin{aligned}\end{aligned}\].

• The vector equation of the plane with normal vector \(\vn =\langle a,b,c \rangle\) containing the points \(P_0 = (x_0, y_0, z_0)\) and \(Q = (x,y,z)\) is \[\begin{aligned}\end{aligned}\].

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.